What are the side effects of having relief pain injection in the left butt in periformis syndrome? How should I convince my doctor to give this injecti

Don't. If your dr isn't comfortable doing this injection there's a reason for it. The side effects are serious and range from organ perforation to abscess formation among others. If you have piriformis syndrome find someone who is experienced in it's management.
Side effects. Infection, bleeding and sciatic nerve injury are the risks of concern. See your doctor. If he examines you and diagnoses piriformis syndrome he may offer the injection.